If the diameter of a sphere is doubled, how does its surface area change? (a) It increases two times (b) It increases three times (c) It increases four times (d) no change
Understand the Problem
The question is asking about the relationship between the diameter of a sphere and its surface area, specifically how the surface area changes when the diameter is doubled. To solve this, we will use the formula for the surface area of a sphere, which is 4πr², where r is the radius. When the diameter is doubled, the radius also doubles, leading us to find how 4π(2r)² compares to the original surface area.
Answer
The new surface area is $4$ times the original surface area.
Answer for screen readers
The new surface area is 4 times the original surface area.
Steps to Solve
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Identify the original surface area The surface area ( A ) of a sphere is given by the formula: $$ A = 4\pi r^2 $$ where ( r ) is the original radius.
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Determine the new radius after doubling the diameter When the diameter is doubled, the new radius ( r' ) is: $$ r' = 2r $$
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Calculate the new surface area with the new radius Using the new radius ( r' ), the new surface area ( A' ) will be: $$ A' = 4\pi (r')^2 $$ Substituting ( r' ): $$ A' = 4\pi (2r)^2 $$
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Simplify the expression for the new surface area Calculating ( (2r)^2 ): $$ (2r)^2 = 4r^2 $$ Thus, the new surface area becomes: $$ A' = 4\pi \cdot 4r^2 = 16\pi r^2 $$
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Compare the new surface area to the original surface area We can now compare the new surface area ( A' ) to the original surface area ( A ): $$ \frac{A'}{A} = \frac{16\pi r^2}{4\pi r^2} = 4 $$ This indicates that the new surface area is 4 times the original surface area.
The new surface area is 4 times the original surface area.
More Information
When you double the diameter of a sphere, the surface area increases significantly due to the quadratic relationship with the radius. This concept is widely applicable in geometry and can help to understand the changes in measurements of three-dimensional shapes.
Tips
- Confusing radius and diameter: Remember to halve the diameter to get the radius before using formulas.
- Forgetting to square the radius when calculating surface area: Always ensure you square the radius before multiplying by ( 4\pi ).
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